3.85 \(\int x^2 (d+i c d x)^3 (a+b \tan ^{-1}(c x))^2 \, dx\)

Optimal. Leaf size=402 \[ -\frac{14 i b^2 d^3 \text{PolyLog}\left (2,1-\frac{2}{1+i c x}\right )}{15 c^3}-\frac{1}{6} i c^3 d^3 x^6 \left (a+b \tan ^{-1}(c x)\right )^2-\frac{3}{5} c^2 d^3 x^5 \left (a+b \tan ^{-1}(c x)\right )^2+\frac{1}{15} i b c^2 d^3 x^5 \left (a+b \tan ^{-1}(c x)\right )+\frac{11 i a b d^3 x}{6 c^2}-\frac{37 i d^3 \left (a+b \tan ^{-1}(c x)\right )^2}{20 c^3}-\frac{28 b d^3 \log \left (\frac{2}{1+i c x}\right ) \left (a+b \tan ^{-1}(c x)\right )}{15 c^3}+\frac{3}{4} i c d^3 x^4 \left (a+b \tan ^{-1}(c x)\right )^2+\frac{3}{10} b c d^3 x^4 \left (a+b \tan ^{-1}(c x)\right )+\frac{1}{3} d^3 x^3 \left (a+b \tan ^{-1}(c x)\right )^2-\frac{11}{18} i b d^3 x^3 \left (a+b \tan ^{-1}(c x)\right )-\frac{14 b d^3 x^2 \left (a+b \tan ^{-1}(c x)\right )}{15 c}-\frac{113 i b^2 d^3 \log \left (c^2 x^2+1\right )}{90 c^3}+\frac{37 b^2 d^3 x}{30 c^2}+\frac{11 i b^2 d^3 x \tan ^{-1}(c x)}{6 c^2}-\frac{37 b^2 d^3 \tan ^{-1}(c x)}{30 c^3}-\frac{1}{60} i b^2 c d^3 x^4+\frac{61 i b^2 d^3 x^2}{180 c}-\frac{1}{10} b^2 d^3 x^3 \]

[Out]

(((11*I)/6)*a*b*d^3*x)/c^2 + (37*b^2*d^3*x)/(30*c^2) + (((61*I)/180)*b^2*d^3*x^2)/c - (b^2*d^3*x^3)/10 - (I/60
)*b^2*c*d^3*x^4 - (37*b^2*d^3*ArcTan[c*x])/(30*c^3) + (((11*I)/6)*b^2*d^3*x*ArcTan[c*x])/c^2 - (14*b*d^3*x^2*(
a + b*ArcTan[c*x]))/(15*c) - ((11*I)/18)*b*d^3*x^3*(a + b*ArcTan[c*x]) + (3*b*c*d^3*x^4*(a + b*ArcTan[c*x]))/1
0 + (I/15)*b*c^2*d^3*x^5*(a + b*ArcTan[c*x]) - (((37*I)/20)*d^3*(a + b*ArcTan[c*x])^2)/c^3 + (d^3*x^3*(a + b*A
rcTan[c*x])^2)/3 + ((3*I)/4)*c*d^3*x^4*(a + b*ArcTan[c*x])^2 - (3*c^2*d^3*x^5*(a + b*ArcTan[c*x])^2)/5 - (I/6)
*c^3*d^3*x^6*(a + b*ArcTan[c*x])^2 - (28*b*d^3*(a + b*ArcTan[c*x])*Log[2/(1 + I*c*x)])/(15*c^3) - (((113*I)/90
)*b^2*d^3*Log[1 + c^2*x^2])/c^3 - (((14*I)/15)*b^2*d^3*PolyLog[2, 1 - 2/(1 + I*c*x)])/c^3

________________________________________________________________________________________

Rubi [A]  time = 1.19617, antiderivative size = 402, normalized size of antiderivative = 1., number of steps used = 52, number of rules used = 15, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.6, Rules used = {4876, 4852, 4916, 321, 203, 4920, 4854, 2402, 2315, 266, 43, 4846, 260, 4884, 302} \[ -\frac{14 i b^2 d^3 \text{PolyLog}\left (2,1-\frac{2}{1+i c x}\right )}{15 c^3}-\frac{1}{6} i c^3 d^3 x^6 \left (a+b \tan ^{-1}(c x)\right )^2-\frac{3}{5} c^2 d^3 x^5 \left (a+b \tan ^{-1}(c x)\right )^2+\frac{1}{15} i b c^2 d^3 x^5 \left (a+b \tan ^{-1}(c x)\right )+\frac{11 i a b d^3 x}{6 c^2}-\frac{37 i d^3 \left (a+b \tan ^{-1}(c x)\right )^2}{20 c^3}-\frac{28 b d^3 \log \left (\frac{2}{1+i c x}\right ) \left (a+b \tan ^{-1}(c x)\right )}{15 c^3}+\frac{3}{4} i c d^3 x^4 \left (a+b \tan ^{-1}(c x)\right )^2+\frac{3}{10} b c d^3 x^4 \left (a+b \tan ^{-1}(c x)\right )+\frac{1}{3} d^3 x^3 \left (a+b \tan ^{-1}(c x)\right )^2-\frac{11}{18} i b d^3 x^3 \left (a+b \tan ^{-1}(c x)\right )-\frac{14 b d^3 x^2 \left (a+b \tan ^{-1}(c x)\right )}{15 c}-\frac{113 i b^2 d^3 \log \left (c^2 x^2+1\right )}{90 c^3}+\frac{37 b^2 d^3 x}{30 c^2}+\frac{11 i b^2 d^3 x \tan ^{-1}(c x)}{6 c^2}-\frac{37 b^2 d^3 \tan ^{-1}(c x)}{30 c^3}-\frac{1}{60} i b^2 c d^3 x^4+\frac{61 i b^2 d^3 x^2}{180 c}-\frac{1}{10} b^2 d^3 x^3 \]

Antiderivative was successfully verified.

[In]

Int[x^2*(d + I*c*d*x)^3*(a + b*ArcTan[c*x])^2,x]

[Out]

(((11*I)/6)*a*b*d^3*x)/c^2 + (37*b^2*d^3*x)/(30*c^2) + (((61*I)/180)*b^2*d^3*x^2)/c - (b^2*d^3*x^3)/10 - (I/60
)*b^2*c*d^3*x^4 - (37*b^2*d^3*ArcTan[c*x])/(30*c^3) + (((11*I)/6)*b^2*d^3*x*ArcTan[c*x])/c^2 - (14*b*d^3*x^2*(
a + b*ArcTan[c*x]))/(15*c) - ((11*I)/18)*b*d^3*x^3*(a + b*ArcTan[c*x]) + (3*b*c*d^3*x^4*(a + b*ArcTan[c*x]))/1
0 + (I/15)*b*c^2*d^3*x^5*(a + b*ArcTan[c*x]) - (((37*I)/20)*d^3*(a + b*ArcTan[c*x])^2)/c^3 + (d^3*x^3*(a + b*A
rcTan[c*x])^2)/3 + ((3*I)/4)*c*d^3*x^4*(a + b*ArcTan[c*x])^2 - (3*c^2*d^3*x^5*(a + b*ArcTan[c*x])^2)/5 - (I/6)
*c^3*d^3*x^6*(a + b*ArcTan[c*x])^2 - (28*b*d^3*(a + b*ArcTan[c*x])*Log[2/(1 + I*c*x)])/(15*c^3) - (((113*I)/90
)*b^2*d^3*Log[1 + c^2*x^2])/c^3 - (((14*I)/15)*b^2*d^3*PolyLog[2, 1 - 2/(1 + I*c*x)])/c^3

Rule 4876

Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)*((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_))^(q_.), x_Symbol] :> Int[Ex
pandIntegrand[(a + b*ArcTan[c*x])^p, (f*x)^m*(d + e*x)^q, x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && IGtQ[p,
0] && IntegerQ[q] && (GtQ[q, 0] || NeQ[a, 0] || IntegerQ[m])

Rule 4852

Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[((d*x)^(m + 1)*(a + b*ArcTa
n[c*x])^p)/(d*(m + 1)), x] - Dist[(b*c*p)/(d*(m + 1)), Int[((d*x)^(m + 1)*(a + b*ArcTan[c*x])^(p - 1))/(1 + c^
2*x^2), x], x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[p, 0] && (EqQ[p, 1] || IntegerQ[m]) && NeQ[m, -1]

Rule 4916

Int[(((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)*((f_.)*(x_))^(m_))/((d_) + (e_.)*(x_)^2), x_Symbol] :> Dist[f^2/
e, Int[(f*x)^(m - 2)*(a + b*ArcTan[c*x])^p, x], x] - Dist[(d*f^2)/e, Int[((f*x)^(m - 2)*(a + b*ArcTan[c*x])^p)
/(d + e*x^2), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && GtQ[p, 0] && GtQ[m, 1]

Rule 321

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(c^(n - 1)*(c*x)^(m - n + 1)*(a + b*x^n
)^(p + 1))/(b*(m + n*p + 1)), x] - Dist[(a*c^n*(m - n + 1))/(b*(m + n*p + 1)), Int[(c*x)^(m - n)*(a + b*x^n)^p
, x], x] /; FreeQ[{a, b, c, p}, x] && IGtQ[n, 0] && GtQ[m, n - 1] && NeQ[m + n*p + 1, 0] && IntBinomialQ[a, b,
c, n, m, p, x]

Rule 203

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTan[(Rt[b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[b, 2]), x] /;
FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a, 0] || GtQ[b, 0])

Rule 4920

Int[(((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)*(x_))/((d_) + (e_.)*(x_)^2), x_Symbol] :> -Simp[(I*(a + b*ArcTan
[c*x])^(p + 1))/(b*e*(p + 1)), x] - Dist[1/(c*d), Int[(a + b*ArcTan[c*x])^p/(I - c*x), x], x] /; FreeQ[{a, b,
c, d, e}, x] && EqQ[e, c^2*d] && IGtQ[p, 0]

Rule 4854

Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)/((d_) + (e_.)*(x_)), x_Symbol] :> -Simp[((a + b*ArcTan[c*x])^p*Lo
g[2/(1 + (e*x)/d)])/e, x] + Dist[(b*c*p)/e, Int[((a + b*ArcTan[c*x])^(p - 1)*Log[2/(1 + (e*x)/d)])/(1 + c^2*x^
2), x], x] /; FreeQ[{a, b, c, d, e}, x] && IGtQ[p, 0] && EqQ[c^2*d^2 + e^2, 0]

Rule 2402

Int[Log[(c_.)/((d_) + (e_.)*(x_))]/((f_) + (g_.)*(x_)^2), x_Symbol] :> -Dist[e/g, Subst[Int[Log[2*d*x]/(1 - 2*
d*x), x], x, 1/(d + e*x)], x] /; FreeQ[{c, d, e, f, g}, x] && EqQ[c, 2*d] && EqQ[e^2*f + d^2*g, 0]

Rule 2315

Int[Log[(c_.)*(x_)]/((d_) + (e_.)*(x_)), x_Symbol] :> -Simp[PolyLog[2, 1 - c*x]/e, x] /; FreeQ[{c, d, e}, x] &
& EqQ[e + c*d, 0]

Rule 266

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a
+ b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rule 4846

Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.), x_Symbol] :> Simp[x*(a + b*ArcTan[c*x])^p, x] - Dist[b*c*p, Int[
(x*(a + b*ArcTan[c*x])^(p - 1))/(1 + c^2*x^2), x], x] /; FreeQ[{a, b, c}, x] && IGtQ[p, 0]

Rule 260

Int[(x_)^(m_.)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> Simp[Log[RemoveContent[a + b*x^n, x]]/(b*n), x] /; FreeQ
[{a, b, m, n}, x] && EqQ[m, n - 1]

Rule 4884

Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)/((d_) + (e_.)*(x_)^2), x_Symbol] :> Simp[(a + b*ArcTan[c*x])^(p +
1)/(b*c*d*(p + 1)), x] /; FreeQ[{a, b, c, d, e, p}, x] && EqQ[e, c^2*d] && NeQ[p, -1]

Rule 302

Int[(x_)^(m_)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> Int[PolynomialDivide[x^m, a + b*x^n, x], x] /; FreeQ[{a,
b}, x] && IGtQ[m, 0] && IGtQ[n, 0] && GtQ[m, 2*n - 1]

Rubi steps

\begin{align*} \int x^2 (d+i c d x)^3 \left (a+b \tan ^{-1}(c x)\right )^2 \, dx &=\int \left (d^3 x^2 \left (a+b \tan ^{-1}(c x)\right )^2+3 i c d^3 x^3 \left (a+b \tan ^{-1}(c x)\right )^2-3 c^2 d^3 x^4 \left (a+b \tan ^{-1}(c x)\right )^2-i c^3 d^3 x^5 \left (a+b \tan ^{-1}(c x)\right )^2\right ) \, dx\\ &=d^3 \int x^2 \left (a+b \tan ^{-1}(c x)\right )^2 \, dx+\left (3 i c d^3\right ) \int x^3 \left (a+b \tan ^{-1}(c x)\right )^2 \, dx-\left (3 c^2 d^3\right ) \int x^4 \left (a+b \tan ^{-1}(c x)\right )^2 \, dx-\left (i c^3 d^3\right ) \int x^5 \left (a+b \tan ^{-1}(c x)\right )^2 \, dx\\ &=\frac{1}{3} d^3 x^3 \left (a+b \tan ^{-1}(c x)\right )^2+\frac{3}{4} i c d^3 x^4 \left (a+b \tan ^{-1}(c x)\right )^2-\frac{3}{5} c^2 d^3 x^5 \left (a+b \tan ^{-1}(c x)\right )^2-\frac{1}{6} i c^3 d^3 x^6 \left (a+b \tan ^{-1}(c x)\right )^2-\frac{1}{3} \left (2 b c d^3\right ) \int \frac{x^3 \left (a+b \tan ^{-1}(c x)\right )}{1+c^2 x^2} \, dx-\frac{1}{2} \left (3 i b c^2 d^3\right ) \int \frac{x^4 \left (a+b \tan ^{-1}(c x)\right )}{1+c^2 x^2} \, dx+\frac{1}{5} \left (6 b c^3 d^3\right ) \int \frac{x^5 \left (a+b \tan ^{-1}(c x)\right )}{1+c^2 x^2} \, dx+\frac{1}{3} \left (i b c^4 d^3\right ) \int \frac{x^6 \left (a+b \tan ^{-1}(c x)\right )}{1+c^2 x^2} \, dx\\ &=\frac{1}{3} d^3 x^3 \left (a+b \tan ^{-1}(c x)\right )^2+\frac{3}{4} i c d^3 x^4 \left (a+b \tan ^{-1}(c x)\right )^2-\frac{3}{5} c^2 d^3 x^5 \left (a+b \tan ^{-1}(c x)\right )^2-\frac{1}{6} i c^3 d^3 x^6 \left (a+b \tan ^{-1}(c x)\right )^2-\frac{1}{2} \left (3 i b d^3\right ) \int x^2 \left (a+b \tan ^{-1}(c x)\right ) \, dx+\frac{1}{2} \left (3 i b d^3\right ) \int \frac{x^2 \left (a+b \tan ^{-1}(c x)\right )}{1+c^2 x^2} \, dx-\frac{\left (2 b d^3\right ) \int x \left (a+b \tan ^{-1}(c x)\right ) \, dx}{3 c}+\frac{\left (2 b d^3\right ) \int \frac{x \left (a+b \tan ^{-1}(c x)\right )}{1+c^2 x^2} \, dx}{3 c}+\frac{1}{5} \left (6 b c d^3\right ) \int x^3 \left (a+b \tan ^{-1}(c x)\right ) \, dx-\frac{1}{5} \left (6 b c d^3\right ) \int \frac{x^3 \left (a+b \tan ^{-1}(c x)\right )}{1+c^2 x^2} \, dx+\frac{1}{3} \left (i b c^2 d^3\right ) \int x^4 \left (a+b \tan ^{-1}(c x)\right ) \, dx-\frac{1}{3} \left (i b c^2 d^3\right ) \int \frac{x^4 \left (a+b \tan ^{-1}(c x)\right )}{1+c^2 x^2} \, dx\\ &=-\frac{b d^3 x^2 \left (a+b \tan ^{-1}(c x)\right )}{3 c}-\frac{1}{2} i b d^3 x^3 \left (a+b \tan ^{-1}(c x)\right )+\frac{3}{10} b c d^3 x^4 \left (a+b \tan ^{-1}(c x)\right )+\frac{1}{15} i b c^2 d^3 x^5 \left (a+b \tan ^{-1}(c x)\right )-\frac{i d^3 \left (a+b \tan ^{-1}(c x)\right )^2}{3 c^3}+\frac{1}{3} d^3 x^3 \left (a+b \tan ^{-1}(c x)\right )^2+\frac{3}{4} i c d^3 x^4 \left (a+b \tan ^{-1}(c x)\right )^2-\frac{3}{5} c^2 d^3 x^5 \left (a+b \tan ^{-1}(c x)\right )^2-\frac{1}{6} i c^3 d^3 x^6 \left (a+b \tan ^{-1}(c x)\right )^2-\frac{1}{3} \left (i b d^3\right ) \int x^2 \left (a+b \tan ^{-1}(c x)\right ) \, dx+\frac{1}{3} \left (i b d^3\right ) \int \frac{x^2 \left (a+b \tan ^{-1}(c x)\right )}{1+c^2 x^2} \, dx+\frac{1}{3} \left (b^2 d^3\right ) \int \frac{x^2}{1+c^2 x^2} \, dx+\frac{\left (3 i b d^3\right ) \int \left (a+b \tan ^{-1}(c x)\right ) \, dx}{2 c^2}-\frac{\left (3 i b d^3\right ) \int \frac{a+b \tan ^{-1}(c x)}{1+c^2 x^2} \, dx}{2 c^2}-\frac{\left (2 b d^3\right ) \int \frac{a+b \tan ^{-1}(c x)}{i-c x} \, dx}{3 c^2}-\frac{\left (6 b d^3\right ) \int x \left (a+b \tan ^{-1}(c x)\right ) \, dx}{5 c}+\frac{\left (6 b d^3\right ) \int \frac{x \left (a+b \tan ^{-1}(c x)\right )}{1+c^2 x^2} \, dx}{5 c}+\frac{1}{2} \left (i b^2 c d^3\right ) \int \frac{x^3}{1+c^2 x^2} \, dx-\frac{1}{10} \left (3 b^2 c^2 d^3\right ) \int \frac{x^4}{1+c^2 x^2} \, dx-\frac{1}{15} \left (i b^2 c^3 d^3\right ) \int \frac{x^5}{1+c^2 x^2} \, dx\\ &=\frac{3 i a b d^3 x}{2 c^2}+\frac{b^2 d^3 x}{3 c^2}-\frac{14 b d^3 x^2 \left (a+b \tan ^{-1}(c x)\right )}{15 c}-\frac{11}{18} i b d^3 x^3 \left (a+b \tan ^{-1}(c x)\right )+\frac{3}{10} b c d^3 x^4 \left (a+b \tan ^{-1}(c x)\right )+\frac{1}{15} i b c^2 d^3 x^5 \left (a+b \tan ^{-1}(c x)\right )-\frac{101 i d^3 \left (a+b \tan ^{-1}(c x)\right )^2}{60 c^3}+\frac{1}{3} d^3 x^3 \left (a+b \tan ^{-1}(c x)\right )^2+\frac{3}{4} i c d^3 x^4 \left (a+b \tan ^{-1}(c x)\right )^2-\frac{3}{5} c^2 d^3 x^5 \left (a+b \tan ^{-1}(c x)\right )^2-\frac{1}{6} i c^3 d^3 x^6 \left (a+b \tan ^{-1}(c x)\right )^2-\frac{2 b d^3 \left (a+b \tan ^{-1}(c x)\right ) \log \left (\frac{2}{1+i c x}\right )}{3 c^3}+\frac{1}{5} \left (3 b^2 d^3\right ) \int \frac{x^2}{1+c^2 x^2} \, dx+\frac{\left (i b d^3\right ) \int \left (a+b \tan ^{-1}(c x)\right ) \, dx}{3 c^2}-\frac{\left (i b d^3\right ) \int \frac{a+b \tan ^{-1}(c x)}{1+c^2 x^2} \, dx}{3 c^2}-\frac{\left (6 b d^3\right ) \int \frac{a+b \tan ^{-1}(c x)}{i-c x} \, dx}{5 c^2}+\frac{\left (3 i b^2 d^3\right ) \int \tan ^{-1}(c x) \, dx}{2 c^2}-\frac{\left (b^2 d^3\right ) \int \frac{1}{1+c^2 x^2} \, dx}{3 c^2}+\frac{\left (2 b^2 d^3\right ) \int \frac{\log \left (\frac{2}{1+i c x}\right )}{1+c^2 x^2} \, dx}{3 c^2}+\frac{1}{9} \left (i b^2 c d^3\right ) \int \frac{x^3}{1+c^2 x^2} \, dx+\frac{1}{4} \left (i b^2 c d^3\right ) \operatorname{Subst}\left (\int \frac{x}{1+c^2 x} \, dx,x,x^2\right )-\frac{1}{10} \left (3 b^2 c^2 d^3\right ) \int \left (-\frac{1}{c^4}+\frac{x^2}{c^2}+\frac{1}{c^4 \left (1+c^2 x^2\right )}\right ) \, dx-\frac{1}{30} \left (i b^2 c^3 d^3\right ) \operatorname{Subst}\left (\int \frac{x^2}{1+c^2 x} \, dx,x,x^2\right )\\ &=\frac{11 i a b d^3 x}{6 c^2}+\frac{37 b^2 d^3 x}{30 c^2}-\frac{1}{10} b^2 d^3 x^3-\frac{b^2 d^3 \tan ^{-1}(c x)}{3 c^3}+\frac{3 i b^2 d^3 x \tan ^{-1}(c x)}{2 c^2}-\frac{14 b d^3 x^2 \left (a+b \tan ^{-1}(c x)\right )}{15 c}-\frac{11}{18} i b d^3 x^3 \left (a+b \tan ^{-1}(c x)\right )+\frac{3}{10} b c d^3 x^4 \left (a+b \tan ^{-1}(c x)\right )+\frac{1}{15} i b c^2 d^3 x^5 \left (a+b \tan ^{-1}(c x)\right )-\frac{37 i d^3 \left (a+b \tan ^{-1}(c x)\right )^2}{20 c^3}+\frac{1}{3} d^3 x^3 \left (a+b \tan ^{-1}(c x)\right )^2+\frac{3}{4} i c d^3 x^4 \left (a+b \tan ^{-1}(c x)\right )^2-\frac{3}{5} c^2 d^3 x^5 \left (a+b \tan ^{-1}(c x)\right )^2-\frac{1}{6} i c^3 d^3 x^6 \left (a+b \tan ^{-1}(c x)\right )^2-\frac{28 b d^3 \left (a+b \tan ^{-1}(c x)\right ) \log \left (\frac{2}{1+i c x}\right )}{15 c^3}-\frac{\left (2 i b^2 d^3\right ) \operatorname{Subst}\left (\int \frac{\log (2 x)}{1-2 x} \, dx,x,\frac{1}{1+i c x}\right )}{3 c^3}+\frac{\left (i b^2 d^3\right ) \int \tan ^{-1}(c x) \, dx}{3 c^2}-\frac{\left (3 b^2 d^3\right ) \int \frac{1}{1+c^2 x^2} \, dx}{10 c^2}-\frac{\left (3 b^2 d^3\right ) \int \frac{1}{1+c^2 x^2} \, dx}{5 c^2}+\frac{\left (6 b^2 d^3\right ) \int \frac{\log \left (\frac{2}{1+i c x}\right )}{1+c^2 x^2} \, dx}{5 c^2}-\frac{\left (3 i b^2 d^3\right ) \int \frac{x}{1+c^2 x^2} \, dx}{2 c}+\frac{1}{18} \left (i b^2 c d^3\right ) \operatorname{Subst}\left (\int \frac{x}{1+c^2 x} \, dx,x,x^2\right )+\frac{1}{4} \left (i b^2 c d^3\right ) \operatorname{Subst}\left (\int \left (\frac{1}{c^2}-\frac{1}{c^2 \left (1+c^2 x\right )}\right ) \, dx,x,x^2\right )-\frac{1}{30} \left (i b^2 c^3 d^3\right ) \operatorname{Subst}\left (\int \left (-\frac{1}{c^4}+\frac{x}{c^2}+\frac{1}{c^4 \left (1+c^2 x\right )}\right ) \, dx,x,x^2\right )\\ &=\frac{11 i a b d^3 x}{6 c^2}+\frac{37 b^2 d^3 x}{30 c^2}+\frac{17 i b^2 d^3 x^2}{60 c}-\frac{1}{10} b^2 d^3 x^3-\frac{1}{60} i b^2 c d^3 x^4-\frac{37 b^2 d^3 \tan ^{-1}(c x)}{30 c^3}+\frac{11 i b^2 d^3 x \tan ^{-1}(c x)}{6 c^2}-\frac{14 b d^3 x^2 \left (a+b \tan ^{-1}(c x)\right )}{15 c}-\frac{11}{18} i b d^3 x^3 \left (a+b \tan ^{-1}(c x)\right )+\frac{3}{10} b c d^3 x^4 \left (a+b \tan ^{-1}(c x)\right )+\frac{1}{15} i b c^2 d^3 x^5 \left (a+b \tan ^{-1}(c x)\right )-\frac{37 i d^3 \left (a+b \tan ^{-1}(c x)\right )^2}{20 c^3}+\frac{1}{3} d^3 x^3 \left (a+b \tan ^{-1}(c x)\right )^2+\frac{3}{4} i c d^3 x^4 \left (a+b \tan ^{-1}(c x)\right )^2-\frac{3}{5} c^2 d^3 x^5 \left (a+b \tan ^{-1}(c x)\right )^2-\frac{1}{6} i c^3 d^3 x^6 \left (a+b \tan ^{-1}(c x)\right )^2-\frac{28 b d^3 \left (a+b \tan ^{-1}(c x)\right ) \log \left (\frac{2}{1+i c x}\right )}{15 c^3}-\frac{31 i b^2 d^3 \log \left (1+c^2 x^2\right )}{30 c^3}-\frac{i b^2 d^3 \text{Li}_2\left (1-\frac{2}{1+i c x}\right )}{3 c^3}-\frac{\left (6 i b^2 d^3\right ) \operatorname{Subst}\left (\int \frac{\log (2 x)}{1-2 x} \, dx,x,\frac{1}{1+i c x}\right )}{5 c^3}-\frac{\left (i b^2 d^3\right ) \int \frac{x}{1+c^2 x^2} \, dx}{3 c}+\frac{1}{18} \left (i b^2 c d^3\right ) \operatorname{Subst}\left (\int \left (\frac{1}{c^2}-\frac{1}{c^2 \left (1+c^2 x\right )}\right ) \, dx,x,x^2\right )\\ &=\frac{11 i a b d^3 x}{6 c^2}+\frac{37 b^2 d^3 x}{30 c^2}+\frac{61 i b^2 d^3 x^2}{180 c}-\frac{1}{10} b^2 d^3 x^3-\frac{1}{60} i b^2 c d^3 x^4-\frac{37 b^2 d^3 \tan ^{-1}(c x)}{30 c^3}+\frac{11 i b^2 d^3 x \tan ^{-1}(c x)}{6 c^2}-\frac{14 b d^3 x^2 \left (a+b \tan ^{-1}(c x)\right )}{15 c}-\frac{11}{18} i b d^3 x^3 \left (a+b \tan ^{-1}(c x)\right )+\frac{3}{10} b c d^3 x^4 \left (a+b \tan ^{-1}(c x)\right )+\frac{1}{15} i b c^2 d^3 x^5 \left (a+b \tan ^{-1}(c x)\right )-\frac{37 i d^3 \left (a+b \tan ^{-1}(c x)\right )^2}{20 c^3}+\frac{1}{3} d^3 x^3 \left (a+b \tan ^{-1}(c x)\right )^2+\frac{3}{4} i c d^3 x^4 \left (a+b \tan ^{-1}(c x)\right )^2-\frac{3}{5} c^2 d^3 x^5 \left (a+b \tan ^{-1}(c x)\right )^2-\frac{1}{6} i c^3 d^3 x^6 \left (a+b \tan ^{-1}(c x)\right )^2-\frac{28 b d^3 \left (a+b \tan ^{-1}(c x)\right ) \log \left (\frac{2}{1+i c x}\right )}{15 c^3}-\frac{113 i b^2 d^3 \log \left (1+c^2 x^2\right )}{90 c^3}-\frac{14 i b^2 d^3 \text{Li}_2\left (1-\frac{2}{1+i c x}\right )}{15 c^3}\\ \end{align*}

Mathematica [A]  time = 1.3733, size = 369, normalized size = 0.92 \[ \frac{d^3 \left (168 i b^2 \text{PolyLog}\left (2,-e^{2 i \tan ^{-1}(c x)}\right )-30 i a^2 c^6 x^6-108 a^2 c^5 x^5+135 i a^2 c^4 x^4+60 a^2 c^3 x^3+12 i a b c^5 x^5+54 a b c^4 x^4-110 i a b c^3 x^3-168 a b c^2 x^2+168 a b \log \left (c^2 x^2+1\right )+2 b \tan ^{-1}(c x) \left (3 a \left (-10 i c^6 x^6-36 c^5 x^5+45 i c^4 x^4+20 c^3 x^3-55 i\right )+b \left (6 i c^5 x^5+27 c^4 x^4-55 i c^3 x^3-84 c^2 x^2+165 i c x-111\right )-168 b \log \left (1+e^{2 i \tan ^{-1}(c x)}\right )\right )+330 i a b c x-162 a b-3 i b^2 c^4 x^4-18 b^2 c^3 x^3+61 i b^2 c^2 x^2-226 i b^2 \log \left (c^2 x^2+1\right )+3 b^2 (c x-i)^4 \left (-10 i c^2 x^2+4 c x+i\right ) \tan ^{-1}(c x)^2+222 b^2 c x+64 i b^2\right )}{180 c^3} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[x^2*(d + I*c*d*x)^3*(a + b*ArcTan[c*x])^2,x]

[Out]

(d^3*(-162*a*b + (64*I)*b^2 + (330*I)*a*b*c*x + 222*b^2*c*x - 168*a*b*c^2*x^2 + (61*I)*b^2*c^2*x^2 + 60*a^2*c^
3*x^3 - (110*I)*a*b*c^3*x^3 - 18*b^2*c^3*x^3 + (135*I)*a^2*c^4*x^4 + 54*a*b*c^4*x^4 - (3*I)*b^2*c^4*x^4 - 108*
a^2*c^5*x^5 + (12*I)*a*b*c^5*x^5 - (30*I)*a^2*c^6*x^6 + 3*b^2*(-I + c*x)^4*(I + 4*c*x - (10*I)*c^2*x^2)*ArcTan
[c*x]^2 + 2*b*ArcTan[c*x]*(b*(-111 + (165*I)*c*x - 84*c^2*x^2 - (55*I)*c^3*x^3 + 27*c^4*x^4 + (6*I)*c^5*x^5) +
3*a*(-55*I + 20*c^3*x^3 + (45*I)*c^4*x^4 - 36*c^5*x^5 - (10*I)*c^6*x^6) - 168*b*Log[1 + E^((2*I)*ArcTan[c*x])
]) + 168*a*b*Log[1 + c^2*x^2] - (226*I)*b^2*Log[1 + c^2*x^2] + (168*I)*b^2*PolyLog[2, -E^((2*I)*ArcTan[c*x])])
)/(180*c^3)

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Maple [B]  time = 0.1, size = 712, normalized size = 1.8 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2*(d+I*c*d*x)^3*(a+b*arctan(c*x))^2,x)

[Out]

-7/15*I/c^3*d^3*b^2*ln(c*x-I)*ln(-1/2*I*(c*x+I))+1/15*I*c^2*d^3*a*b*x^5+7/15*I/c^3*d^3*b^2*ln(c*x-I)*ln(c^2*x^
2+1)-6/5*c^2*d^3*a*b*arctan(c*x)*x^5+1/15*I*c^2*d^3*b^2*arctan(c*x)*x^5-1/6*I*c^3*d^3*b^2*arctan(c*x)^2*x^6+3/
4*I*c*d^3*b^2*arctan(c*x)^2*x^4+7/15*I/c^3*d^3*b^2*ln(c*x+I)*ln(1/2*I*(c*x-I))-11/6*I/c^3*d^3*a*b*arctan(c*x)-
7/15*I/c^3*d^3*b^2*ln(c*x+I)*ln(c^2*x^2+1)+1/3*d^3*b^2*arctan(c*x)^2*x^3-3/5*c^2*d^3*a^2*x^5+3/2*I*c*d^3*a*b*a
rctan(c*x)*x^4-1/3*I*c^3*d^3*a*b*arctan(c*x)*x^6+2/3*d^3*a*b*arctan(c*x)*x^3-14/15/c*d^3*b^2*arctan(c*x)*x^2-3
/5*c^2*d^3*b^2*arctan(c*x)^2*x^5+14/15/c^3*d^3*b^2*arctan(c*x)*ln(c^2*x^2+1)-11/18*I*d^3*a*b*x^3-1/6*I*c^3*d^3
*a^2*x^6+3/4*I*c*d^3*a^2*x^4+7/30*I/c^3*d^3*b^2*ln(c*x+I)^2+7/15*I/c^3*d^3*b^2*dilog(1/2*I*(c*x-I))-7/15*I/c^3
*d^3*b^2*dilog(-1/2*I*(c*x+I))+3/10*c*d^3*a*b*x^4-14/15/c*d^3*a*b*x^2+3/10*c*d^3*b^2*arctan(c*x)*x^4-11/18*I*d
^3*b^2*arctan(c*x)*x^3-11/12*I/c^3*d^3*b^2*arctan(c*x)^2-7/30*I/c^3*d^3*b^2*ln(c*x-I)^2+14/15/c^3*d^3*a*b*ln(c
^2*x^2+1)+1/3*d^3*a^2*x^3-1/10*b^2*d^3*x^3+37/30*b^2*d^3*x/c^2-37/30*b^2*d^3*arctan(c*x)/c^3+11/6*I*a*b*d^3*x/
c^2+61/180*I*b^2*d^3*x^2/c+11/6*I*b^2*d^3*x*arctan(c*x)/c^2-1/60*I*b^2*c*d^3*x^4-113/90*I*b^2*d^3*ln(c^2*x^2+1
)/c^3

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*(d+I*c*d*x)^3*(a+b*arctan(c*x))^2,x, algorithm="maxima")

[Out]

-1/6*I*a^2*c^3*d^3*x^6 - 3/5*a^2*c^2*d^3*x^5 + 3/4*I*a^2*c*d^3*x^4 - 1/45*I*(15*x^6*arctan(c*x) - c*((3*c^4*x^
5 - 5*c^2*x^3 + 15*x)/c^6 - 15*arctan(c*x)/c^7))*a*b*c^3*d^3 - 3/10*(4*x^5*arctan(c*x) - c*((c^2*x^4 - 2*x^2)/
c^4 + 2*log(c^2*x^2 + 1)/c^6))*a*b*c^2*d^3 + 1/3*a^2*d^3*x^3 + 1/2*I*(3*x^4*arctan(c*x) - c*((c^2*x^3 - 3*x)/c
^4 + 3*arctan(c*x)/c^5))*a*b*c*d^3 + 1/3*(2*x^3*arctan(c*x) - c*(x^2/c^2 - log(c^2*x^2 + 1)/c^4))*a*b*d^3 - 1/
960*(40*I*b^2*c^3*d^3*x^6 + 144*b^2*c^2*d^3*x^5 - 180*I*b^2*c*d^3*x^4 - 80*b^2*d^3*x^3)*arctan(c*x)^2 + 1/960*
(40*b^2*c^3*d^3*x^6 - 144*I*b^2*c^2*d^3*x^5 - 180*b^2*c*d^3*x^4 + 80*I*b^2*d^3*x^3)*arctan(c*x)*log(c^2*x^2 +
1) - 1/960*(-10*I*b^2*c^3*d^3*x^6 - 36*b^2*c^2*d^3*x^5 + 45*I*b^2*c*d^3*x^4 + 20*b^2*d^3*x^3)*log(c^2*x^2 + 1)
^2 - I*integrate(1/240*(180*(b^2*c^5*d^3*x^7 - 2*b^2*c^3*d^3*x^5 - 3*b^2*c*d^3*x^3)*arctan(c*x)^2 + 15*(b^2*c^
5*d^3*x^7 - 2*b^2*c^3*d^3*x^5 - 3*b^2*c*d^3*x^3)*log(c^2*x^2 + 1)^2 - 2*(46*b^2*c^4*d^3*x^6 - 65*b^2*c^2*d^3*x
^4)*arctan(c*x) + (10*b^2*c^5*d^3*x^7 - 81*b^2*c^3*d^3*x^5 + 20*b^2*c*d^3*x^3 - 60*(3*b^2*c^4*d^3*x^6 + 2*b^2*
c^2*d^3*x^4 - b^2*d^3*x^2)*arctan(c*x))*log(c^2*x^2 + 1))/(c^2*x^2 + 1), x) - integrate(1/240*(180*(3*b^2*c^4*
d^3*x^6 + 2*b^2*c^2*d^3*x^4 - b^2*d^3*x^2)*arctan(c*x)^2 + 15*(3*b^2*c^4*d^3*x^6 + 2*b^2*c^2*d^3*x^4 - b^2*d^3
*x^2)*log(c^2*x^2 + 1)^2 + 2*(10*b^2*c^5*d^3*x^7 - 81*b^2*c^3*d^3*x^5 + 20*b^2*c*d^3*x^3)*arctan(c*x) + (46*b^
2*c^4*d^3*x^6 - 65*b^2*c^2*d^3*x^4 + 60*(b^2*c^5*d^3*x^7 - 2*b^2*c^3*d^3*x^5 - 3*b^2*c*d^3*x^3)*arctan(c*x))*l
og(c^2*x^2 + 1))/(c^2*x^2 + 1), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{1}{240} \,{\left (10 i \, b^{2} c^{3} d^{3} x^{6} + 36 \, b^{2} c^{2} d^{3} x^{5} - 45 i \, b^{2} c d^{3} x^{4} - 20 \, b^{2} d^{3} x^{3}\right )} \log \left (-\frac{c x + i}{c x - i}\right )^{2} +{\rm integral}\left (\frac{-60 i \, a^{2} c^{5} d^{3} x^{7} - 180 \, a^{2} c^{4} d^{3} x^{6} + 120 i \, a^{2} c^{3} d^{3} x^{5} - 120 \, a^{2} c^{2} d^{3} x^{4} + 180 i \, a^{2} c d^{3} x^{3} + 60 \, a^{2} d^{3} x^{2} +{\left (60 \, a b c^{5} d^{3} x^{7} +{\left (-180 i \, a b - 10 \, b^{2}\right )} c^{4} d^{3} x^{6} - 12 \,{\left (10 \, a b - 3 i \, b^{2}\right )} c^{3} d^{3} x^{5} +{\left (-120 i \, a b + 45 \, b^{2}\right )} c^{2} d^{3} x^{4} - 20 \,{\left (9 \, a b + i \, b^{2}\right )} c d^{3} x^{3} + 60 i \, a b d^{3} x^{2}\right )} \log \left (-\frac{c x + i}{c x - i}\right )}{60 \,{\left (c^{2} x^{2} + 1\right )}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*(d+I*c*d*x)^3*(a+b*arctan(c*x))^2,x, algorithm="fricas")

[Out]

1/240*(10*I*b^2*c^3*d^3*x^6 + 36*b^2*c^2*d^3*x^5 - 45*I*b^2*c*d^3*x^4 - 20*b^2*d^3*x^3)*log(-(c*x + I)/(c*x -
I))^2 + integral(1/60*(-60*I*a^2*c^5*d^3*x^7 - 180*a^2*c^4*d^3*x^6 + 120*I*a^2*c^3*d^3*x^5 - 120*a^2*c^2*d^3*x
^4 + 180*I*a^2*c*d^3*x^3 + 60*a^2*d^3*x^2 + (60*a*b*c^5*d^3*x^7 + (-180*I*a*b - 10*b^2)*c^4*d^3*x^6 - 12*(10*a
*b - 3*I*b^2)*c^3*d^3*x^5 + (-120*I*a*b + 45*b^2)*c^2*d^3*x^4 - 20*(9*a*b + I*b^2)*c*d^3*x^3 + 60*I*a*b*d^3*x^
2)*log(-(c*x + I)/(c*x - I)))/(c^2*x^2 + 1), x)

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Sympy [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: AttributeError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**2*(d+I*c*d*x)**3*(a+b*atan(c*x))**2,x)

[Out]

Exception raised: AttributeError

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (i \, c d x + d\right )}^{3}{\left (b \arctan \left (c x\right ) + a\right )}^{2} x^{2}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*(d+I*c*d*x)^3*(a+b*arctan(c*x))^2,x, algorithm="giac")

[Out]

integrate((I*c*d*x + d)^3*(b*arctan(c*x) + a)^2*x^2, x)